For every complex number $z$, $|z+1|\ge{1\over{\sqrt2}}$ or $|z^2+1|\ge1$: correct or incorrect?

Question

I recently started complex numbers but I think the following question is half correct and half incorrect.

Prove that for any complex number $z$
$$|z+1|\ge{1\over{\sqrt2}}\qquad\text{or}\qquad|z^2+1|\ge1$$

I think the first inequality is wrong. For eg. if we take $z=-1-{\iota\over2}$, $|z+1|$ will be equal to $1\over2$.

Is my thinking correct or wrong?

Answer 1

You are indeed proving that the first inequality does not hold for all complex numbers.

The point, however, is that you have to prove that for every complex number $z$, one of the inequalities holds.

In your example, you have to check what $z^2+1$ is: it is $3/4+i$, the norm of which is greater than $1$.